Calculus Examples

\int e^{- 4 x} d x

$\int e^{- 4 x} d x$

Step 1

Let

u = - 4 x

$u = - 4 x$ . Then

d u = - 4 d x

$d u = - 4 d x$ , so

- \frac{1}{4} d u = d x

$- \frac{1}{4} d u = d x$ . Rewrite using

u

$u$ and

d

$d$

u

$u$ .

Tap for more steps...

Step 1.1

Let

u = - 4 x

$u = - 4 x$ . Find

\frac{d u}{d x}

$\frac{d u}{d x}$ .

Tap for more steps...

Step 1.1.1

Differentiate

- 4 x

$- 4 x$ .

\frac{d}{d x} [- 4 x]

$\frac{d}{d x} [- 4 x]$

Step 1.1.2

Since

- 4

$- 4$ is constant with respect to

x

$x$ , the derivative of

- 4 x

$- 4 x$ with respect to

x

$x$ is

- 4 \frac{d}{d x} [x]

$- 4 \frac{d}{d x} [x]$ .

- 4 \frac{d}{d x} [x]

$- 4 \frac{d}{d x} [x]$

Step 1.1.3

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 1

$n = 1$ .

- 4 \cdot 1

$- 4 \cdot 1$

Step 1.1.4

Multiply

- 4

$- 4$ by

1

$1$ .

- 4

$- 4$

- 4

$- 4$

Step 1.2

Rewrite the problem using

u

$u$ and

d u

$d u$ .

\int e^{u} \frac{1}{- 4} d u

$\int e^{u} \frac{1}{- 4} d u$

\int e^{u} \frac{1}{- 4} d u

$\int e^{u} \frac{1}{- 4} d u$

Step 2

Simplify.

Tap for more steps...

Step 2.1

Move the negative in front of the fraction.

\int e^{u} (- \frac{1}{4}) d u

$\int e^{u} (- \frac{1}{4}) d u$

Step 2.2

Combine

e^{u}

$e^{u}$ and

\frac{1}{4}

$\frac{1}{4}$ .

\int - \frac{e^{u}}{4} d u

$\int - \frac{e^{u}}{4} d u$

\int - \frac{e^{u}}{4} d u

$\int - \frac{e^{u}}{4} d u$

Step 3

Since

- 1

$- 1$ is constant with respect to

u

$u$ , move

- 1

$- 1$ out of the integral.

- \int \frac{e^{u}}{4} d u

$- \int \frac{e^{u}}{4} d u$

Step 4

Since

\frac{1}{4}

$\frac{1}{4}$ is constant with respect to

u

$u$ , move

\frac{1}{4}

$\frac{1}{4}$ out of the integral.

- (\frac{1}{4} \int e^{u} d u)

$- (\frac{1}{4} \int e^{u} d u)$

Step 5

The integral of

e^{u}

$e^{u}$ with respect to

u

$u$ is

e^{u}

$e^{u}$ .

- \frac{1}{4} (e^{u} + C)

$- \frac{1}{4} (e^{u} + C)$

Step 6

Simplify.

- \frac{1}{4} e^{u} + C

$- \frac{1}{4} e^{u} + C$

Step 7

Replace all occurrences of

u

$u$ with

- 4 x

$- 4 x$ .

- \frac{1}{4} e^{- 4 x} + C

$- \frac{1}{4} e^{- 4 x} + C$

$\int_{}^{} e^{- 4 x}$

(

(

)

)

|

|

[

[

]

]

√

√





≥

≥





∫

∫













7

7

8

8

9

9













≤

≤





°

°

θ

θ









4

4

5

5

6

6

/

/

^

^

×

×

>

>





π

π













1

1

2

2

3

3

-

-

+

+

÷

÷

<

<





∞

∞





!

!



,

,

0

0

.

.

%

%



=

=









$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$